Remark 3.10 (Net flux) 
To show the existence of gradient-flow solutions in the sense of null-minimizers of \(\mathcal {I}_n\)  we will have to jump from gradient-flow solutions in the sense of [1]}, see Theorem REF . The expressions for net-fluxes are in fact contractions of those for one-way or uni-directional fluxes, as discussed in Section , which we use to show that the two notions of gradient-flow solutions are equivalent.

